"추상대수학"의 두 판 사이의 차이

수학노트
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1번째 줄: 1번째 줄:
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<h5>간단한 요약</h5>
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==개요==
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* 현대대수학의 기본적인 언어이자 대상인, 군, 환, 체의 기본적인 용어 및 이론을 공부함.
 
* 현대대수학의 기본적인 언어이자 대상인, 군, 환, 체의 기본적인 용어 및 이론을 공부함.
 
* 갈루아 이론 - 군론을 통해 확장체 혹은 대수방정식의 해가 가진 대칭성을 들여다 봄.
 
* 갈루아 이론 - 군론을 통해 확장체 혹은 대수방정식의 해가 가진 대칭성을 들여다 봄.
  
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<h5>선수 과목 또는 알고 있으면 좋은 것들</h5>
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==선수 과목 또는 알고 있으면 좋은 것들==
  
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*  고교 수준의 대수학<br>
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*  고교 수준의 대수학
 
** 다항식, 다항방정식
 
** 다항식, 다항방정식
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*  기초적인 선형대수학<br>
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*  기초적인 선형대수학
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**  기저, 차원, 선형사상, 행렬, 행렬식<br>
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**  기저, 차원, 선형사상, 행렬, 행렬식
  
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<h5>다루는 대상</h5>
 
  
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*  군(group)<br>
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==다루는 대상==
 +
 
 +
*  군(group)
 
** 대칭성을 기술하는 언어
 
** 대칭성을 기술하는 언어
 
** 항등원, 역원,
 
** 항등원, 역원,
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*  환(ring)<br>
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*  환(ring)
 
** 덧셈, 뺄셈, 곱하기가 가능하며, 덧셈과 곱셈 사이에 분배법칙이 성립.
 
** 덧셈, 뺄셈, 곱하기가 가능하며, 덧셈과 곱셈 사이에 분배법칙이 성립.
 
** 정수의 집합, 다항식의 집합, n x n 행렬들의 집합
 
** 정수의 집합, 다항식의 집합, n x n 행렬들의 집합
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*  체(field)<br>
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*  체(field)
 
** 실수, 복소수와 같이 사칙연산이 가능.
 
** 실수, 복소수와 같이 사칙연산이 가능.
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**  좀더 일반적으로 곱셈의 교환법칙을 가정하지 않는 경우는 division ring이라 부름.<br>  <br>
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**  좀더 일반적으로 곱셈의 교환법칙을 가정하지 않는 경우는 division ring이라 부름.
  
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<h5>중요한 개념 및 정리</h5>
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==중요한 개념 및 정리==
  
 
* [[순환군]]
 
* [[순환군]]
 
* [[군론(group theory)|군론]]
 
* [[군론(group theory)|군론]]
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* 체론(field theory)
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* [[유한생성 아벨군의 기본정리]]
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* 유한생성 아벨군의 기본정리
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* [[#|체론(field theory)]]
 
* ideal
 
* ideal
 
* 유한체
 
* 유한체
 
* 갈루아 체확장
 
* 갈루아 체확장
  
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<h5>유명한 정리 혹은 생각할만한 문제</h5>
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==유명한 정리 혹은 생각할만한 문제==
  
 
* [[대수학의 기본정리]](The fundamental theorem of algebras)의 대수적 증명은 가능한가?
 
* [[대수학의 기본정리]](The fundamental theorem of algebras)의 대수적 증명은 가능한가?
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* [[해밀턴의 사원수(quarternions)|해밀턴의 사원수]]<br>
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* [[해밀턴의 사원수(quarternions)|해밀턴의 사원수]]
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**  아래 참고할만한 자료<br>
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**  아래 참고할만한 자료
 
*** [http://www.jstor.org/stable/2315349 The Impossibility of a Division Algebra of Vectors in Three Dimensional Space]
 
*** [http://www.jstor.org/stable/2315349 The Impossibility of a Division Algebra of Vectors in Three Dimensional Space]
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*** [http://www.jstor.org/stable/2689449 Hamilton's Discovery of Quaternions]<br>
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*** [http://www.jstor.org/stable/2689449 Hamilton's Discovery of Quaternions]
 
* [[가우스와 정17각형의 작도]]
 
* [[가우스와 정17각형의 작도]]
 
* [[그리스 3대 작도 불가능문제]]를 군론을 통해 해결할 수 있음.
 
* [[그리스 3대 작도 불가능문제]]를 군론을 통해 해결할 수 있음.
 
* [[5차방정식과 근의 공식|일반적인 5차 이상의 방정식의 대수적 해가 존재하지 않음에 대한 아벨의 증명]]
 
* [[5차방정식과 근의 공식|일반적인 5차 이상의 방정식의 대수적 해가 존재하지 않음에 대한 아벨의 증명]]
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*  유클리드 도메인이 아닌 PID<br>
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*  유클리드 도메인이 아닌 PID
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**  아래 참고할만한 자료<br>
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**  아래 참고할만한 자료
 
*** [http://www.jstor.org/stable/2322908 A Principal Ideal Domain That Is Not a Euclidean Domain]
 
*** [http://www.jstor.org/stable/2322908 A Principal Ideal Domain That Is Not a Euclidean Domain]
 
* [[7개의 프리즈 패턴]]
 
* [[7개의 프리즈 패턴]]
 
* [[17 Plane Crystallographic groups]]
 
* [[17 Plane Crystallographic groups]]
  
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<h5>다른 과목과의 관련성</h5>
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==다른 과목과의 관련성==
  
 
* [[초등정수론|정수론]]
 
* [[초등정수론|정수론]]
 
* [[선형대수학]]
 
* [[선형대수학]]
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* [[리만곡면론|대수곡선론]]<br>
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* [[리만곡면론|대수곡선론]]
 
** 대수기하학 입문으로서의 대수곡선론
 
** 대수기하학 입문으로서의 대수곡선론
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* [[대수적위상수학]]<br>
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* [[대수적위상수학]]
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**  군론<br>
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**  군론
 
*** fundamental group을 정의하기 위해 필요
 
*** fundamental group을 정의하기 위해 필요
 
*** covering space의 deck transformation group
 
*** covering space의 deck transformation group
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**  유한생성 아벨군의 기본정리<br>
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**  유한생성 아벨군의 기본정리
 
*** 호몰로지를 이해하기 위해 필요
 
*** 호몰로지를 이해하기 위해 필요
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*  조합론<br>
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*  조합론
 
** 번사이드 보조정리
 
** 번사이드 보조정리
  
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+
  
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<h5>관련된 대학원 과목 또는 더 공부하면 좋은 것들</h5>
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==관련된 대학원 과목 또는 더 공부하면 좋은 것들==
  
 
* 펠릭스 클라인의 '정이십면체와 5차방정식'
 
* 펠릭스 클라인의 '정이십면체와 5차방정식'
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*  semisimple rings<br>
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*  semisimple rings
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** Artin–Wedderburn theorem
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** [[아틴-웨더번 정리(Artin–Wedderburn theorem)]]
 
* 유한군의 표현론
 
* 유한군의 표현론
 
* [[대수적수론|대수적정수론]]
 
* [[대수적수론|대수적정수론]]
 
* [[Classical groups]]
 
* [[Classical groups]]
  
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<h5>표준적인 교과서</h5>
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==매스매티카 파일 및 계산 리소스==
 +
* http://homepage.math.uiowa.edu/~goodman/22m121.dir/2006/22m121.html
 +
 
 +
 
 +
==표준적인 교과서==
 +
 
 +
* John B. Fraleigh [http://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/0201763907 A First Course in Abstract Algebra]
 +
 
 +
 
 +
==관련도서==
 +
 
 +
* Israel Kleiner [http://www.amazon.com/exec/obidos/ASIN/0817646841/ebooksclub-20/ A History of Abstract Algebra]
 +
 
 +
  
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* [http://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/0201763907 A First Course in Abstract Algebra]<br>
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==관련논문==
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** John B. Fraleigh
 
  
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<h5>추천도서 및 보조교재</h5>
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* I. G. Bashmakova and A. N. Rudakov [http://www.jstor.org/stable/2975015 The Evolution of Algebra 1800-1870] , <cite>The American Mathematical Monthly</cite>, Vol. 102, No. 3 (Mar., 1995), pp. 266-270
 +
* [http://www.jstor.org/stable/2690312 The Evolution of Group Theory: A Brief Survey]
 +
** Israel Kleiner, <cite>Mathematics Magazine</cite>, Vol. 59, No. 4 (Oct., 1986), pp. 195-215
 +
* [http://www.jstor.org/stable/2690624 A History of Lagrange's Theorem on Groups]
 +
** Richard L. Roth, <cite>Mathematics Magazine</cite>, Vol. 74, No. 2 (Apr., 2001), pp. 99-108                             
 +
* [http://www.jstor.org/stable/2689449 Hamilton's Discovery of Quaternions]
 +
** B. L. van der Waerden, <cite>Mathematics Magazine</cite>, Vol. 49, No. 5 (Nov., 1976), pp. 227-234
 +
* [http://www.jstor.org/stable/2974935 The Genesis of the Abstract Ring Concept]
 +
** Israel Kleiner, <cite>The American Mathematical Monthly</cite>, Vol. 103, No. 5 (May, 1996), pp. 417-424
 +
* [http://www.jstor.org/stable/2691011 A Historically Focused Course in Abstract Algebra]
 +
** Israel Kleiner, <cite>Mathematics Magazine</cite>, Vol. 71, No. 2 (Apr., 1998), pp. 105-111
 +
* [http://www.jstor.org/stable/2325119 Galois Theory for Beginners]
 +
** John Stillwell, <cite>The American Mathematical Monthly</cite>, Vol. 101, No. 1 (Jan., 1994), pp. 22-27
 +
* [http://www.jstor.org/stable/2974763 Niels Hendrik Abel and Equations of the Fifth Degree]
 +
** Michael I. Rosen, <cite>The American Mathematical Monthly</cite>, Vol. 102, No. 6 (Jun. - Jul., 1995), pp. 495-505
 +
* [http://www.jstor.org/stable/2322908 A Principal Ideal Domain That Is Not a Euclidean Domain]
 +
** Oscar A. Campoli, <cite>The American Mathematical Monthly</cite>, Vol. 95, No. 9 (Nov., 1988), pp. 868-871
 +
* [http://www.jstor.org/stable/2974984 Principal Ideal Domains are Almost Euclidean]
 +
** John Greene, <cite>The American Mathematical Monthly</cite>, Vol. 104, No. 2 (Feb., 1997), pp. 154-156
 +
[[분류:교과목]]
 +
[[분류:추상대수학]]
  
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* [http://www.amazon.com/exec/obidos/ASIN/0817646841/ebooksclub-20/ A History of Abstract Algebra]<br>
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== 노트 ==
−
**  Israel Kleiner<br>
 
  
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+
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q159943 Q159943]
 +
===말뭉치===
 +
# Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner.<ref name="ref_32c6abca">[http://abstract.ups.edu/ Abstract Algebra: Theory and Applications (A Free Textbook)]</ref>
 +
# Five of them will be taking college courses in differential equations, abstract algebra and discrete mathematics as 10th-graders at Pasadena High School this fall.<ref name="ref_cfa4eaf5">[https://www.merriam-webster.com/dictionary/abstract%20algebra Definition of Abstract Algebra by Merriam-Webster]</ref>
 +
# At the height of her mathematical powers, doing new work on abstract algebra, Noether died after an operation on an ovarian cyst.<ref name="ref_cfa4eaf5" />
 +
# If the operations satisfy familiar arithmetic rules (such as associativity, commutativity, and distributivity) the set will have a particularly “rich” algebraic structure.<ref name="ref_fd0599a4">[https://www.britannica.com/science/modern-algebra Modern algebra | mathematics]</ref>
 +
# Sets with the richest algebraic structure are known as fields.<ref name="ref_fd0599a4" />
 +
# In fact, finite fields motivated the early development of abstract algebra.<ref name="ref_fd0599a4" />
 +
# Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems.<ref name="ref_53cb43d7">[https://mathworld.wolfram.com/AbstractAlgebra.html Abstract Algebra -- from Wolfram MathWorld]</ref>
 +
# discrete mathematics are sometimes considered branches of abstract algebra.<ref name="ref_53cb43d7" />
 +
# In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.<ref name="ref_f9c755ac">[https://en.wikipedia.org/wiki/Abstract_algebra Abstract algebra]</ref>
 +
# Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras.<ref name="ref_f9c755ac" />
 +
# Algebraic structures, with their associated homomorphisms, form mathematical categories.<ref name="ref_f9c755ac" />
 +
# As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra.<ref name="ref_f9c755ac" />
 +
# But the richness of abstract algebra comes from the idea that you can use abstractions of a concept that are easy to understand to explain more complex behavior!<ref name="ref_9d850f01">[https://math.stackexchange.com/questions/855828/what-is-abstract-algebra-essentially What is Abstract Algebra essentially?]</ref>
 +
# This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra.<ref name="ref_84fa5ed8">[http://www.math.niu.edu/~beachy/aaol/contents.html ABSTRACT ALGEBRA ON LINE: Contents]</ref>
 +
# It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course.<ref name="ref_84fa5ed8" />
 +
# Algebra has also played a significant role in clarifying and highlighting notions of logic, at the core of exact philosophy for millennia.<ref name="ref_6874edf0">[https://plato.stanford.edu/entries/algebra/ Algebra (Stanford Encyclopedia of Philosophy)]</ref>
 +
# A number of branches of mathematics have found algebra such an effective tool that they have spawned algebraic subbranches.<ref name="ref_6874edf0" />
 +
# Groups, rings and fields only scratch the surface of abstract algebra.<ref name="ref_6874edf0" />
 +
# Boolean algebras abstract the algebra of sets.<ref name="ref_6874edf0" />
 +
# "Abstract Algebra" is a clearly written, self-contained basic algebra text for graduate students, with a generous amount of additional material that suggests the scope of contemporary algebra.<ref name="ref_a30cf7c2">[https://www.springer.com/gp/book/9780387715674 Pierre Antoine Grillet]</ref>
 +
# The last chapters, on universal algebras and categories, including tripleability, give valuable general views of algebra.<ref name="ref_a30cf7c2" />
 +
# This book is on abstract algebra (abstract algebraic systems), an advanced set of topics related to algebra, including groups, rings, ideals, fields, and more.<ref name="ref_cbc73e9b">[https://en.wikibooks.org/wiki/Abstract_Algebra Wikibooks, open books for an open world]</ref>
 +
# Primarily, we will follow Wikibooks' Abstract Algebra textbook.<ref name="ref_7cd593b6">[https://en.wikiversity.org/wiki/Introduction_to_Abstract_Algebra Introduction to Abstract Algebra]</ref>
 +
# A good textbook to pick up is Topics in Algebra by I.N. Herstein.<ref name="ref_7cd593b6" />
 +
# Shed the societal and cultural narratives holding you back and let step-by-step A First Course in Abstract Algebra textbook solutions reorient your old paradigms.<ref name="ref_17f28728">[https://www.slader.com/textbook/9780201763904-a-first-course-in-abstract-algebra-7th-edition/ Solutions to A First Course in Abstract Algebra (9780201763904) :: Homework Help and Answers :: Slader]</ref>
 +
# Unlock your A First Course in Abstract Algebra PDF (Profound Dynamic Fulfillment) today.<ref name="ref_17f28728" />
 +
# This course develops in the theme of "Arithmetic congruence, and abstract algebraic structures.<ref name="ref_ceda1da5">[https://math.gatech.edu/courses/math/4107 Georgia Institute of Technology]</ref>
 +
# This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences.<ref name="ref_8424066c">[https://www.degruyter.com/view/title/319702?language=en Abstract Algebra]</ref>
 +
# Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world.<ref name="ref_2eef2cc6">[https://www.cambridge.org/core_title/gb/491143 Abstract algebra applications]</ref>
 +
# Perhaps no other subject of undergraduate mathematics is as challenging to learn (and to teach) as abstract algebra.<ref name="ref_2eef2cc6" />
 +
# Many of the applications of modern algebra are explained in a thoughtful way that will help motivate students to study abstract concepts.<ref name="ref_2eef2cc6" />
 +
# Abstract Algebra by Dummit & Foote is a standard textbook used by colleges and universities.<ref name="ref_99735e06">[https://www.socratica.com/subject/abstract-algebra Abstract Algebra]</ref>
 +
# It covers all the topics for a solid first course in Abstract Algebra.<ref name="ref_99735e06" />
 +
# Considerable emphasis is placed on the algebraic system consisting of the congruence classes mod n under the usual operations of addition and multiplication.<ref name="ref_c51bd267">[https://www.worldscientific.com/worldscibooks/10.1142/9853 Abstract Algebra]</ref>
 +
# Noether went into research and more or less invented the field of abstract algebra.<ref name="ref_92aa58b2">[https://cosmosmagazine.com/physics/woman-who-invented-abstract-algebra/ The inventor of abstract algebra]</ref>
 +
# The main aim of the course is to introduce you to basic concepts from abstract algebra, especially the notion of a group.<ref name="ref_eea5599c">[https://www.kcl.ac.uk/study/courses-data/modules/4/Introduction-To-Abstract-Algebra-4ccm121a Introduction To Abstract Algebra]</ref>
 +
# The course will help prepare you for further study in abstract algebra as well as familiarize you with tools essential in many other areas of mathematics.<ref name="ref_eea5599c" />
 +
# Computational Problems in Abstract Algebra provides information pertinent to the application of computers to abstract algebra.<ref name="ref_213c6e05">[https://www.sciencedirect.com/book/9780080129754/computational-problems-in-abstract-algebra Computational Problems in Abstract Algebra]</ref>
 +
# The final chapter deals with the computational problems related to invariant factors in linear algebra.<ref name="ref_213c6e05" />
 +
# Mathematicians as well as students of algebra will find this book useful.<ref name="ref_213c6e05" />
 +
===소스===
 +
<references />
  
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<h5 class="parseasinTitle">참고할만한 자료</h5>
+
== 메타데이터 ==
  
−
* [http://www.jstor.org/stable/2975015 The Evolution of Algebra 1800-1870]<br>
+
===위키데이터===
−
** I. G. Bashmakova and A. N. Rudakov
+
* ID : [https://www.wikidata.org/wiki/Q159943 Q159943]
−
** <cite>The American Mathematical Monthly</cite>, Vol. 102, No. 3 (Mar., 1995), pp. 266-270
+
===Spacy 패턴 목록===
−
* [http://www.jstor.org/stable/2690312 The Evolution of Group Theory: A Brief Survey]<br>
+
* [{'LOWER': 'abstract'}, {'LEMMA': 'algebra'}]
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** Israel Kleiner
+
* [{'LOWER': 'modern'}, {'LEMMA': 'algebra'}]
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** <cite>Mathematics Magazine</cite>, Vol. 59, No. 4 (Oct., 1986), pp. 195-215
+
* [{'LEMMA': 'algebra'}]
−
* [http://www.jstor.org/stable/2690624 A History of Lagrange's Theorem on Groups]<br>
+
* [{'LEMMA': 'algebraic'}]
−
** Richard L. Roth
 
−
** <cite>Mathematics Magazine</cite>, Vol. 74, No. 2 (Apr., 2001), pp. 99-108                              
 
−
* [http://www.jstor.org/stable/2589500 Field Theory: From Equations to Axiomatization, Part I]<br>
 
−
** Israel Kleiner
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 106, No. 7 (Aug. - Sep., 1999), pp. 677-684
 
−
* [http://www.jstor.org/stable/2589621 Field Theory: From Equations to Axiomatization, Part II]<br>
 
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** Israel Kleiner
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 106, No. 9 (Nov., 1999), pp. 859-863
 
−
* [http://www.jstor.org/stable/4146920 The Arithmetic of Algebraic Numbers: An Elementary Approach]<br>
 
−
** Chi-Kwong Li and David Lutzer
 
−
** <cite>The College Mathematics Journal</cite>, Vol. 35, No. 4 (Sep., 2004), pp. 307-309
 
−
* [http://www.jstor.org/stable/2689449 Hamilton's Discovery of Quaternions]<br>
 
−
** B. L. van der Waerden
 
−
** <cite>Mathematics Magazine</cite>, Vol. 49, No. 5 (Nov., 1976), pp. 227-234
 
−
* [http://www.jstor.org/stable/2315349 The Impossibility of a Division Algebra of Vectors in Three Dimensional Space]<br>
 
−
** Kenneth O. May
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 73, No. 3 (Mar., 1966), pp. 289-291
 
−
* [http://www.jstor.org/stable/2974935 The Genesis of the Abstract Ring Concept]<br>
 
−
** Israel Kleiner
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 103, No. 5 (May, 1996), pp. 417-424
 
−
* [http://www.jstor.org/stable/2691011 A Historically Focused Course in Abstract Algebra]<br>
 
−
** Israel Kleiner
 
−
** <cite>Mathematics Magazine</cite>, Vol. 71, No. 2 (Apr., 1998), pp. 105-111
 
−
* [http://www.jstor.org/stable/2321249 Finite Simple Groups]<br>
 
−
** James F. Hurley and Arunas Rudvalis
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 84, No. 9 (Nov., 1977), pp. 693-714
 
−
* [http://www.jstor.org/stable/2690115 The Search for Finite Simple Groups]<br>
 
−
** Joseph A. Gallian
 
−
** <cite>Mathematics Magazine</cite>, Vol. 49, No. 4 (Sep., 1976), pp. 163-180
 
−
* [http://www.jstor.org/stable/2325119 Galois Theory for Beginners]<br>
 
−
** John Stillwell
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 101, No. 1 (Jan., 1994), pp. 22-27
 
−
* [http://www.jstor.org/stable/2974763 Niels Hendrik Abel and Equations of the Fifth Degree]<br>
 
−
** Michael I. Rosen
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 102, No. 6 (Jun. - Jul., 1995), pp. 495-505
 
−
* [http://www.jstor.org/stable/2975607 What Are Algebraic Integers and What Are They For?]<br>
 
−
** John Stillwell
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 101, No. 3 (Mar., 1994), pp. 266-270
 
−
* [http://www.jstor.org/stable/2322908 A Principal Ideal Domain That Is Not a Euclidean Domain]<br>
 
−
** Oscar A. Campoli
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 95, No. 9 (Nov., 1988), pp. 868-871
 
−
* [http://www.jstor.org/stable/2974984 Principal Ideal Domains are Almost Euclidean]<br>
 
−
** John Greene
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 104, No. 2 (Feb., 1997), pp. 154-156
 

2021년 2월 17일 (수) 06:01 기준 최신판

개요

  • 현대대수학의 기본적인 언어이자 대상인, 군, 환, 체의 기본적인 용어 및 이론을 공부함.
  • 갈루아 이론 - 군론을 통해 확장체 혹은 대수방정식의 해가 가진 대칭성을 들여다 봄.


선수 과목 또는 알고 있으면 좋은 것들

  • 고교 수준의 대수학
    • 다항식, 다항방정식
  • 기초적인 선형대수학
    • 기저, 차원, 선형사상, 행렬, 행렬식


다루는 대상

  • 군(group)
    • 대칭성을 기술하는 언어
    • 항등원, 역원,
  • 환(ring)
    • 덧셈, 뺄셈, 곱하기가 가능하며, 덧셈과 곱셈 사이에 분배법칙이 성립.
    • 정수의 집합, 다항식의 집합, n x n 행렬들의 집합
  • 체(field)
    • 실수, 복소수와 같이 사칙연산이 가능.
    • 좀더 일반적으로 곱셈의 교환법칙을 가정하지 않는 경우는 division ring이라 부름.


중요한 개념 및 정리


유명한 정리 혹은 생각할만한 문제


다른 과목과의 관련성

  • 정수론
  • 선형대수학
  • 대수곡선론
    • 대수기하학 입문으로서의 대수곡선론
  • 대수적위상수학
    • 군론
      • fundamental group을 정의하기 위해 필요
      • covering space의 deck transformation group
    • 유한생성 아벨군의 기본정리
      • 호몰로지를 이해하기 위해 필요
  • 조합론
    • 번사이드 보조정리



관련된 대학원 과목 또는 더 공부하면 좋은 것들


매스매티카 파일 및 계산 리소스


표준적인 교과서


관련도서


관련논문

노트

위키데이터

말뭉치

  1. Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner.[1]
  2. Five of them will be taking college courses in differential equations, abstract algebra and discrete mathematics as 10th-graders at Pasadena High School this fall.[2]
  3. At the height of her mathematical powers, doing new work on abstract algebra, Noether died after an operation on an ovarian cyst.[2]
  4. If the operations satisfy familiar arithmetic rules (such as associativity, commutativity, and distributivity) the set will have a particularly “rich” algebraic structure.[3]
  5. Sets with the richest algebraic structure are known as fields.[3]
  6. In fact, finite fields motivated the early development of abstract algebra.[3]
  7. Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems.[4]
  8. discrete mathematics are sometimes considered branches of abstract algebra.[4]
  9. In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.[5]
  10. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras.[5]
  11. Algebraic structures, with their associated homomorphisms, form mathematical categories.[5]
  12. As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra.[5]
  13. But the richness of abstract algebra comes from the idea that you can use abstractions of a concept that are easy to understand to explain more complex behavior![6]
  14. This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra.[7]
  15. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course.[7]
  16. Algebra has also played a significant role in clarifying and highlighting notions of logic, at the core of exact philosophy for millennia.[8]
  17. A number of branches of mathematics have found algebra such an effective tool that they have spawned algebraic subbranches.[8]
  18. Groups, rings and fields only scratch the surface of abstract algebra.[8]
  19. Boolean algebras abstract the algebra of sets.[8]
  20. "Abstract Algebra" is a clearly written, self-contained basic algebra text for graduate students, with a generous amount of additional material that suggests the scope of contemporary algebra.[9]
  21. The last chapters, on universal algebras and categories, including tripleability, give valuable general views of algebra.[9]
  22. This book is on abstract algebra (abstract algebraic systems), an advanced set of topics related to algebra, including groups, rings, ideals, fields, and more.[10]
  23. Primarily, we will follow Wikibooks' Abstract Algebra textbook.[11]
  24. A good textbook to pick up is Topics in Algebra by I.N. Herstein.[11]
  25. Shed the societal and cultural narratives holding you back and let step-by-step A First Course in Abstract Algebra textbook solutions reorient your old paradigms.[12]
  26. Unlock your A First Course in Abstract Algebra PDF (Profound Dynamic Fulfillment) today.[12]
  27. This course develops in the theme of "Arithmetic congruence, and abstract algebraic structures.[13]
  28. This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences.[14]
  29. Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world.[15]
  30. Perhaps no other subject of undergraduate mathematics is as challenging to learn (and to teach) as abstract algebra.[15]
  31. Many of the applications of modern algebra are explained in a thoughtful way that will help motivate students to study abstract concepts.[15]
  32. Abstract Algebra by Dummit & Foote is a standard textbook used by colleges and universities.[16]
  33. It covers all the topics for a solid first course in Abstract Algebra.[16]
  34. Considerable emphasis is placed on the algebraic system consisting of the congruence classes mod n under the usual operations of addition and multiplication.[17]
  35. Noether went into research and more or less invented the field of abstract algebra.[18]
  36. The main aim of the course is to introduce you to basic concepts from abstract algebra, especially the notion of a group.[19]
  37. The course will help prepare you for further study in abstract algebra as well as familiarize you with tools essential in many other areas of mathematics.[19]
  38. Computational Problems in Abstract Algebra provides information pertinent to the application of computers to abstract algebra.[20]
  39. The final chapter deals with the computational problems related to invariant factors in linear algebra.[20]
  40. Mathematicians as well as students of algebra will find this book useful.[20]

소스

  1. ↑ Abstract Algebra: Theory and Applications (A Free Textbook)
  2. ↑ 2.0 2.1 Definition of Abstract Algebra by Merriam-Webster
  3. ↑ 3.0 3.1 3.2 Modern algebra | mathematics
  4. ↑ 4.0 4.1 Abstract Algebra -- from Wolfram MathWorld
  5. ↑ 5.0 5.1 5.2 5.3 Abstract algebra
  6. ↑ What is Abstract Algebra essentially?
  7. ↑ 7.0 7.1 ABSTRACT ALGEBRA ON LINE: Contents
  8. ↑ 8.0 8.1 8.2 8.3 Algebra (Stanford Encyclopedia of Philosophy)
  9. ↑ 9.0 9.1 Pierre Antoine Grillet
  10. ↑ Wikibooks, open books for an open world
  11. ↑ 11.0 11.1 Introduction to Abstract Algebra
  12. ↑ 12.0 12.1 Solutions to A First Course in Abstract Algebra (9780201763904) :: Homework Help and Answers :: Slader
  13. ↑ Georgia Institute of Technology
  14. ↑ Abstract Algebra
  15. ↑ 15.0 15.1 15.2 Abstract algebra applications
  16. ↑ 16.0 16.1 Abstract Algebra
  17. ↑ Abstract Algebra
  18. ↑ The inventor of abstract algebra
  19. ↑ 19.0 19.1 Introduction To Abstract Algebra
  20. ↑ 20.0 20.1 20.2 Computational Problems in Abstract Algebra

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'abstract'}, {'LEMMA': 'algebra'}]
  • [{'LOWER': 'modern'}, {'LEMMA': 'algebra'}]
  • [{'LEMMA': 'algebra'}]
  • [{'LEMMA': 'algebraic'}]
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